Problems & Exercises

The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to ${\text{cm}}^3\text{/s}$. (b) What is this rate in ${\text{m}}^3\text{/s}$?

Blood is flowing through an artery of radius 2 mm at a rate of 40 cm/s. Determine the flow rate and the volume that passes through the artery in a period of 30 s.

A major artery with a cross-sectional area of $1\text{.}\text{00}{\text{cm}}^2$ branches into 18 smaller arteries, each with an average cross-sectional area of $0\text{.}\text{400}{\text{cm}}^2$. By what factor is the average velocity of the blood reduced when it passes into these branches?

The human circulation system has approximately $1\times {\text{10}}^9$ capillary vessels. Each vessel has a diameter of about $8\mu \text{m}$. Assuming cardiac output is 5 L/min, determine the average velocity of blood flow through each capillary vessel.

The flow rate of blood through a $2\text{.}\text{00}\times {\text{10}}^{\text{–6}}\text{-m}$ -radius capillary is $3\text{.}\text{80}\times {\text{10}}^{-9}{\text{cm}}^3\text{/s}$. (a) What is the speed of the blood flow? (This small speed allows time for diffusion of materials to and from the blood.) (b) Assuming all the blood in the body passes through capillaries, how many of them must there be to carry a total flow of $90\text{.}0{\text{cm}}^3\text{/s}$? (The large number obtained is an overestimate, but it is still reasonable.)

The main uptake air duct of a forced air gas heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 min? The inside volume of the house is equivalent to a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high.

Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)

Unreasonable Results

A mountain stream is 10.0 m wide and averages 2.00 m in depth. During the spring runoff, the flow in the stream reaches $\text{100,000}{\text{m}}^3\text{/s}$. (a) What is the average velocity of the stream under these conditions? (b) What is unreasonable about this velocity? (c) What is unreasonable or inconsistent about the premises?

Suppose you have a wind speed gauge like the pitot tube shown in Figure 12.7(b). By what factor must wind speed increase to double the value of $h$ in the manometer? Is this independent of the moving fluid and the fluid in the manometer?

Calculate the maximum height to which water could be squirted with the hose in Example 12.2 example if it: (a) Emerges from the nozzle. (b) Emerges with the nozzle removed, assuming the same flow rate.

(a) Calculate the approximate force on a square meter of sail, given the horizontal velocity of the wind is 6.00 m/s parallel to its front surface and 3.50 m/s along its back surface. Take the density of air to be $\text{1.29 kg}{\text{/m}}^3$. (The calculation, based on Bernoulli’s principle, is approximate due to the effects of turbulence.) (b) Discuss whether this force is great enough to be effective for propelling a sailboat.

(a) Using Bernoulli’s equation, show that the measured fluid speed $v$ for a pitot tube, like the one in Figure 12.7(b), is given by $v={\left(\frac{2\rho \prime \mathrm{gh}}{\rho }\right)}^{1/2},$ where $h$ is the height of the manometer fluid, $\rho \prime$ is the density of the manometer fluid, $\rho$ is the density of the moving fluid, and $g$ is the acceleration due to gravity. (Note that $v$ is indeed proportional to the square root of $h$, as stated in the text.) (b) Calculate $v$ for moving air if a mercury manometer’s $h$ is 0.200 m.

A frequently quoted rule of thumb in aircraft design is that wings should produce about 1000 N of lift per square meter of wing. (The fact that a wing has a top and bottom surface does not double its area.) (a) At takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the bottom of the wing is 60.0 m/s. Given the sea level density of air to be $1\text{.}\text{29}{\text{kg/m}}^3$, how fast must it move over the upper surface to create the ideal lift? (b) How fast must air move over the upper surface at a cruising speed of 245 m/s and at an altitude where air density is one-fourth that at sea level? (Note that this is not all of the aircraft’s lift—some comes from the body of the plane, some from engine thrust, and so on. Furthermore, Bernoulli’s principle gives an approximate answer because flow over the wing creates turbulence.)

A sump pump (used to drain water from the basement of houses built below the water table) is draining a flooded basement at the rate of 0.750 L/s, with an output pressure of $3.00\times {\text{10}}^5{\text{N/m}}^2$. (a) The water enters a hose with a 3.00-cm inside diameter and rises 2.50 m above the pump. What is its pressure at this point? (b) The hose goes over the foundation wall, losing 0.500 m in height, and widens to 4.00 cm in diameter. What is the pressure now? You may neglect frictional losses in both parts of the problem.

What force is needed to pull one microscope slide over another at a speed of 1.00 cm/s, if there is a 0.500-mm-thick layer of $\text{20º C}$ water between them and the contact area is $8.00{\text{cm}}^2$?

The pressure drop along a length of artery is 100 Pa, the radius is 10 mm, and the flow is laminar. The average speed of the blood is 15 mm/s. (a) What is the net force on the blood in this section of artery? (b) What is the power expended maintaining the flow?

Fluid originally flows through a tube at a rate of $\text{100}{\text{cm}}^3\text{/s}$. To illustrate the sensitivity of flow rate to various factors, calculate the new flow rate for the following changes with all other factors remaining the same as in the original conditions. (a) Pressure difference increases by a factor of 1.50. (b) A new fluid with 3.00 times greater viscosity is substituted. (c) The tube is replaced by one having 4.00 times the length. (d) Another tube is used with a radius 0.100 times the original. (e) Yet another tube is substituted with a radius 0.100 times the original and half the length, and the pressure difference is increased by a factor of 1.50.

Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 10?

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, $F_s=\text{6}\pi r\eta v$. Show that the terminal speed is given by $v=\frac{{2R}^{2}g}{9\eta }\left({\rho }_{\text{s}}-{\rho }_1\right),$ where $R$ is the radius of the sphere, ${\rho }_{\text{s}}$ is its density, and ${\rho }_1$ is the density of the fluid and $\eta$ the coefficient of viscosity.

A skydiver will reach a terminal velocity when the air drag equals their weight. For a skydiver with high speed and a large body, turbulence is a factor. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be $F_{\text{D}}=\frac{1}{2}\rho {\mathrm{Av}}^{\text{2}}$ and setting this equal to the person’s weight, find the terminal speed for a person falling “spread eagle.” Find both a formula and a number for $v_{\text{t}}$, with assumptions as to size.

(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant, as stated in the text. (b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?

When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?

Water supplied to a house by a water main has a pressure of $3\text{.}\text{00}\times {\text{10}}^5{\text{N/m}}^2$ early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose. (a) What pressure is now being supplied to the house, assuming resistance is constant? (b) By what factor did the flow rate in the water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is $5\text{.}\text{00}\times {\text{10}}^5{\text{N/m}}^2$, and the original flow rate was 200 L/min. (c) How many more users are there, assuming each would consume 20.0 L/min in the morning?

Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200.0 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is $8\text{.}\text{00}\times {\text{10}}^6{\text{N/m}}^2$. (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? (c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.

Construct Your Own Problem

Consider a coronary artery constricted by arteriosclerosis. Construct a problem in which you calculate the amount by which the diameter of the artery is decreased, based on an assessment of the decrease in flow rate.

Consider a river that spreads out in a delta region on its way to the sea. Construct a problem in which you calculate the average speed at which water moves in the delta region, based on the speed at which it was moving up river. Among the things to consider are the size and flow rate of the river before it spreads out and its size once it has spread out. You can construct the problem for the river spreading out into one large river or into multiple smaller rivers.

Show that the Reynolds number $N_{\text{R}}$ is unitless by substituting units for all the quantities in its definition and cancelling.

A fire hose has an inside diameter of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of $1\text{.}\text{62}\times {\text{10}}^6{\text{N/m}}^2$. The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Calculate the Reynolds numbers for flow in the fire hose and nozzle to show that the flow in each must be turbulent.

At what flow rate might turbulence begin to develop in a water main with a 0.200-m diameter? Assume a $\text{20º C}$ temperature.

In Take-Home Experiment: Inhalation, we measured the average flow rate $Q$ of air traveling through the trachea during each inhalation. Now calculate the average air speed in meters per second through your trachea during each inhalation. The radius of the trachea in adult humans is approximately ${\text{10}}^{-2}\text{m}$. From the data above, calculate the Reynolds number for the air flow in the trachea during inhalation. Do you expect the air flow to be laminar or turbulent?

Assuming that blood is an ideal fluid, calculate the critical flow rate at which turbulence is a certainty in the aorta. Take the diameter of the aorta to be 2.50 cm. (Turbulence will actually occur at lower average flow rates, because blood is not an ideal fluid. Furthermore, since blood flow pulses, turbulence may occur during only the high-velocity part of each heartbeat.)

What is the ratio of the average distances that oxygen will diffuse in a given time in air and water? Why is this distance less in water (equivalently, why is $D$ less in water)?

(a) Find the average time required for an oxygen molecule to diffuse through a 0.200-mm-thick tear layer on the cornea. (b) How much time is required to diffuse $0\text{.500}{\text{cm}}^3$ of oxygen to the cornea if its surface area is $1\text{.}\text{00}{\text{cm}}^2$?